Divide ABCDE into triangles by diagonals drawn from A. Planes passed through OA and these diagonals will divide the pyramid into triangular pyramids, O-ABC, O-ACD, and O-ADE.

The volume of each triangular pyramid is one-third the product of its base and the common altitude OZ. 8703

Therefore the volume of the whole pyramid is one-third the sum of the bases of the triangular pyramids, i. e., the base of the whole pyramid multiplied by the common altitude.

Q. E. D.

705. COR. I. Pyramids having equivalent bases and equal altitudes are equivalent.

706. COR. II. Any two pyramids are to each other as the products of their bases and altitudes.

707. COR. III. Two pyramids having equivalent bases are to each other as their altitudes.

708. COR. IV. Two pyramids having equal altitudes are to each other as their bases.

709. Two tetraedrons which have a triedral angle of one equal to a triedral angle of the other are to each other as the products of the three edges about the equal triedral angles.

GIVEN the tetraedrons TABC and T'A'B'C' having the triedral angle A in common. Let V and V' denote their respective volumes.

From T and T' let fall the perpendiculars TD and T'D' upon the plane ABC.

The three points A, D, and D' lie in one straight line.

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Now, considering ABC and AB'C' to be the bases of the tetraedrons,

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And since TD is parallel to T'D',

the triangles ATD and AT'D' are similar.

AT
X

АВХАСХАТ

=

V' AB'XAC' AT AB' XAC' XAT'

Q. E. D.

710. Def.-The altitude of a frustum of a pyramid is the perpendicular distance between the planes of its bases.

711. A frustum of a triangular pyramid is equivalent to the sum of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum.

GIVEN the frustum ABC-DEF of a triangular pyramid.

TO PROVE it is equivalent to the sum of three pyramids, etc.

Pass a plane through F, A, C, and another through F, D, C, thus dividing the frustum into three triangular pyramids, F-ABC, C-DEF, and F-DAC.

Call these pyramids P, Q, and R respectively, and represent ABC by B, DEF by b, and the altitude of the frustum by h. It is evident that if B and b be taken as the bases of P and Q, they have the common altitude h.

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Hence Phx B, and Q-hxb.

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It remains to prove that R is equivalent to a pyramid whose altitude is h and whose base is √Bxb.

The pyramids P and R, regarded as having the common vertex and their bases in the same plane, have the same altitude.

Hence

P ABF

R ADF

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But the triangles ABF and ADF have the same altitude, that of the trapezoid ABFD.

Hence R= √P×Q=√}h×B×}h×b=}h× √ B×b. Therefore R is equivalent to a pyramid whose altitude is

h and whose base is √Bxb.

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Q. E. D.

712. Remark.—If we denote the volume of the frustum by V, the proposition may be expressed in the form

V=}h(B+b+√B×b).

Question.-Does it follow from Proposition XXV. that R is a pyramid whose altitude is h and base VB x b?

713. A frustum of any pyramid is equivalent to the sum of three pyramids whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum.

GIVEN

the frustum AC' of the pyramid V-ABCD.

Denote its lower and upper bases by B and b respectively, its altitude by h, and its volume by V.

TO PROVE

Let T-PKL be a triangular pyramid whose base is in the same plane as ABCD and equivalent to ABCD, whose vertex T is on the same side of this plane as and whose altitude is equal to that of V-ABCD.

Prolong the plane of A'B'C'D' to cut T-PKL in the section P'K'L'.

Set B', b', h', V' for the lower base, upper base, altitude, and volume respectively of the triangular frustum PL.

Then

V' = {h' (B' + b' + √B'xb'). (1)

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